Discrete breathers at the interface between a diatomic and a monoatomic granular chain

In the present work, we develop a systematic examination of the existence, stability, and dynamical properties of a discrete breather at the interface between a diatomic and a monoatomic granular chain. We remarkably find that such an “interface breather” is more robust than its bulk diatomic counterpart throughout the gap of the linear spectrum. The latter linear spectral gap needs to exist for the breather state to arise and the relevant spectral conditions are discussed. We illustrate the minimal excitation conditions under which such an interface breather can be “nucleated” and analyze its apparently weak interaction with regular highly nonlinear solitary waveforms.

Figure 1

The schematic set up for the bead configuration: steel-aluminum-$\dots$-steel-aluminum-brass-brass-$\dots$–brass.

Figure 2

(Color online) In the left panel, the eigenfrequencies of the configuration shown in Fig. 1 are shown for 80 beads. The right four panels show the first, second, third, and fourth optical eigenmodes. For clarity, steel beads are shown as gray circles, aluminum beads are shown as blue diamonds, and brass beads are shown as brown squares.

Figure 3

(Color online) Examples of four solutions. The left two are located at the interface, the right two are in the bulk; the top two are closer to the optical band edge, while the bottom two are deeper within the spectral gap, and thus exhibit a more pronounced localization. For clarity, steel beads are shown as gray circles, aluminum beads are shown as blue diamonds, and brass beads are shown as brown squares.

Figure 4

Top: the magnitudes of the Floquet multipliers for the bulk (left) and interface (right) breathers are shown as a function of frequency. In both cases, the real multipliers are shown in black, while the complex multipliers are shown in gray. Bottom: the energies of the bulk (left) and interface (right) breathers are shown as a function of frequency. Note that the vertical scales are the same in each row, and the horizontal scales are the same in each column.

Figure 5

(Color online) Top: the magnitude of the Floquet multipliers for the double (left) and triple (right) breathers is shown as a function of frequency. In both cases, the real multipliers are shown in black, while the complex multipliers are shown in gray. Bottom: example profiles of the double (left) and triple (right) breathers for a frequency of 18.3 kHz. For clarity, steel beads are shown as gray circles, aluminum beads are shown as blue diamonds, and brass beads are shown as brown squares.

Figure 6

(Color online) Results for a continuation of a DB located one unit cell from the interface. Top: the magnitude of the Floquet multipliers for each DB is shown as a function of frequency. The purely real multipliers are highlighted in black. Middle: the energy of each DB is shown as a function of frequency. Bottom: a sample DB solution from this family is shown. Note that for visual clarity, brass is shown in brown (squares), steel is shown in gray (circles), and aluminum is shown in blue (diamonds).

Figure 7

(Color online) Top left: a spacetime plot of the local energy density for a compression pulse passing through the interface. There is a barely visible reflection amounting to about 2% of the total energy. The lower left panel shows a different view of the same simulation. Here, each line represents the total energy in a certain part of the chain. The dashed line (blue) represents the first half of the chain, the solid line (red) represents the interface region, and the dashed-dotted line (green) represents the second half of the chain. There is no significant amount of energy trapped in the interface. The right two panels show a similar picture, except that in this case, there is an interface breather for the pulse to interact with. The pulse causes the breather to shift by 1 cell. Note that the time scale is the same for the top and bottom panels, and the color scale for the top panels is truncated at 1/20 the maximum, in order for the reflections to be visible.

Figure 8

(Color online) Top: the destruction of a bulk breather due to interaction with a compression pulse. Bottom: the destruction of an interface breather due to the oscillatory instabilities.

Figure 9

(Color online) The frequency of the induced breather (or linear mode) when the middle site (to the left of the interface) was excited with a negative displacement is shown as a function of the magnitude of the initial excitation. Recall that $\Delta$ is the amount of overlap between the beads at equilibrium. The dashed lines represent the acoustic and optical band edges.

Figure 10

(Color online) Top left: the space-time graph showing the energy density for a chain of 1000 beads in which the 500th bead was excited to about $1.5\Delta$. After some initial radiation, a DB is excited at the interface. Top right: a zoom of the top left panel, showing that there is indeed a DB located at the interface. Bottom: the profile of the excited breather is shown in conjunction with the exact breather for the corresponding frequency.